Lower bounds on the redundancy of linear codes with disjoint repair groups.

An error correcting code exhibits the t-Disjoint Repair Group Property (t-DRGP) (for message symbols) if it is possible to recover a single symbol of a codeword (message) in t ways, each from a disjoint set of symbols of the codeword. Codes with the DRGP have found applications in private information retrieval (PIR) and distributed storage, and are related to several notions of locality in coding theory. In this work we prove an impossibility result for codes with the DRGP. We show that the redundancy of any code with the t-DRGP is ${{\Omega }}(\sqrt n )$ for all t ≥ 2. Our bound is tight, even including the leading constant, for t = 2, and is tight up to a constant factor for t = O(1). We also show an analogous result for binary codes with the t-DRGP for message symbols, which has applications to PIR.These results first appeared in 2016 and were never published. As our results have not yet been improved upon, and have been referenced by multiple works over the years, we are prompted to publish them now. We hope that publishing these results now will spur more work in the area, and in particular will lead to improved bounds.